I first investigated how the electrode design affects the EDL in the two electrochemical cells. The surface charge density applied on both electrochemical
near the oppositely charged surface in both electrodes, but yielding diffuse EDLs in the bare cell and compact EDLs in the composite cell. The thickness of EDL can be quantified from the charge density profiles of NaCl shown in Figure 4.2C. Within the bare cell (black line) the thickness (i.e., the distance between the electrode and the end of EDL) of the diffuse EDLs near the negative and positive electrodes are ~13.75Å and 14.25Å, respectively. In contrast, within the composite cell (red line) the thickness of the compact layers near the negative and positive electrodes are 4.75Å and 5.25Å, respectively. These data indicate the formation of compact layers at conditions at which diffuse layers are expected.
Because of the presence of the graphene membranes, when the ions migrate from the central pore to the side pores they remain trapped within the side pores instead of returning to the central pore consequently to thermal motion (as observed in the bare cell, see Figure 4.3A). The ability of the graphene membranes to keep the ions within the side pores is a crucial feature of my design. If the membrane holes diameter D is too large, the ions can easily diffuse back from the side pores to the central pore (Figure 4.3B). As a side note, I point out that, because of confinement effects, there are 6 and 4 water molecules in the first hydration shell of Cl- and Na+ ions, respectively, as opposed to 7 and 6 in bulk aqueous solutions [121, 137]. Clearly, the water molecules in the first hydration shells rearrange in a plane parallel to the pore surfaces (Figure 4.3C). Because of this re-arrangement, the EDL thickness found in my composite cell is not only much smaller than that of the diffuse layer but also smaller than the thickness of a typical Helmholtz EDL in which the ions remain hydrated [105, 106].
Figure 4.2 Representative simulation snapshots at equilibrium for the bare (A) and composite (B) electrochemical cells. These simulations are conducted at the surface charge density 3.2µC/cm2. The colour code is the same as that of Figure 4.1. Note that while in the bare cell the ions distribute throughout the entire pore, with counter-ions accumulating near the charged surfaces, in the composite cell the ions are not present within the central pore. Instead they accumulate within the side pores, with Na+ ions near the negatively charged surface, and Cl- ions near the positively charged one. Charge density profiles obtained at equilibrium for bare (black) and composite (red) cells (C). The density distributions are consistent with the formation of diffuse EDLs in the bare cell, as expected at the low salinity and low charge density considered in these simulations. In contrast, the high intensity and narrow peaks observed in the composite cell indicate the formation of compact layers centred at 3.5Å and 3.75Å near negatively and positively charged electrodes, respectively.
Figure 4.3 Z coordinates of representative Na+ and Cl- ions obtained during equilibrium simulations conducted within the composite cell as a function of time when the hole diameter D=10Å (A) and D=15Å (B). The results show that in the
The latter observation has an important practical consequence as reducing the Helmholtz layer thickness enhances the electrode capacitance (C =εA
d [113, 116],
where A, d, and ε are surface area, Helmholtz layer thickness, and dielectric constant, respectively). From the charge density profiles I integrate twice (see Appendix C) the Poisson equation [111, 120] d2ψ(z)
dz2 = −
ρ(z) ε0
(ε0, ψ(z) , and ρ(z)
are vacuum permittivity, electrical potential, and charge density, respectively) to extract the electrical potential profiles near the charged surfaces (Figure 4.4). To conduct this integration I require two boundary conditions. As first condition, I impose that the electric field in correspondence to the center of the pore is zero (!"!" = 0 𝑎𝑡 𝑧 = 𝑅/2, where R is the pore width). The physical reason underpinning this boundary condition is that at the pore center the electric field emitted from the negative electrode neutralizes the one emitted from the positive electrode. As the second boundary condition, I impose that the electrical potential is zero at z = 0. This condition is arbitrary, and it will not affect the potential difference across the EDL. The electrical potential profiles I obtain allow us to calculate the capacitance using the equation C = σ
ψ , where σ is the surface charge density and ψ is the potential drop across each EDL (see Table 4.1).
The capacitance predicted for the composite electrode is much larger than that predicted for the bare electrode, as expected due to the change in the EDL thickness. It is perhaps more important to point out that the capacitance predicted for the composite electrode considered in my simulations is ~70-80% those reported for electrochemical cells that employ ionic liquids [119, 138, 139].
Figure 4.4 Electrical potential profiles as a function of distance Z between two electrodes. The potential drop across the EDL is the difference between the potential found at the interface and that determined at the end of the EDL. These results are obtained by integrating the charge density profiles twice using the Poisson equation following the procedure described in the Appendix C.
Table 4.1 EDL thickness, potential drop across the EDL (up to the EDL thickness), and capacitance obtained for bare and composite electrodes.
Negatively-charged electrode Positively-charged electrode
dEDL (Å) Potential drop (V) Capacitance µF/cm2 dEDL (Å) Potential drop (V) Capacitance µF/cm2 Composite 4.75 0.72 4.44 5.25 0.81 3.95 Bare 13.75 1.71 1.87 14.25 1.99 1.60
To study the effect of the salinity on the performance of the composite cell, I provide in Figure 4.5 the number of NaCl pairs accumulated within the side-pores as a function of the total number of NaCl pairs initially present in the salty water. These equilibrium simulations were conducted at surface charge density of 3.2µC/cm2. The
the central pore. This indicate that increasing the system salinity above 5.45g/l does not affect the number of ions in the side pores, and hence the capacitance of the Helmholtz EDL.
Figure 4.5 Number of NaCl pairs accumulated within the side-pores as a function of the total number of NaCl pairs initially present in the salty water.
In Figure 4.6 I compare the results obtained from the simulations of the composite cell when the side-pore width H is 7Å and 10Å. In these simulations, the surface charge density is 4.2µC/cm2 and salinity is α~9g/l. Visual inspection of the simulation snapshot (panel A and B) indicates that when H=10Å there is two water layers inside the side-pore. As a results the Helmholtz EDL thickness near the negatively charged electrode (panel C) increases from dHelmholtz=4.25Å when H=7Å
(red line) to dHelmholtz=6.75Å when H=10Å (black line) (see Table 4.2 for more
details on dHelmholtz near positively charged electrode). Because the EDL thickness
increases the capacitance of the composite electrodes decreases. σ =
Figure 4.6 The effect of side-pore size H on the capacitance of the composite electrode. Representative simulation snapshots at equilibrium for side-pore size H=7Å (panel A) and H=10Å (panel B). The colour code is the same as that used in Figure 4.1. Charge density profiles (panel C) obtained at equilibrium for the composite electrochemical cells in which the side-pore size H=7Å (red) and H=10Å (black). Electrical potential profiles (panel D) across the composite electrode when H=7Å (red) and H=10Å (black). See Table 4.2 for thickness, potential drops, and capacitance of EDLs.
Table 4.2 Thickness, potential drop, and capacitance estimated for composite electrodes with side-pore size H=7Å and H=10Å. The correspondent simulation results are summarized in Figure 4.6
Negatively charged electrode Positively charged electrode
dHelmholtz Potential drop Capacitance dHelmholtz Potential drop Capacitance
H=7Å 4.25Å 1V 4.2µF/cm2 5.25Å 1.1V 3.8µF/cm2
are initiated either from the last configuration obtained from the equilibrium simulations (i.e., the ions are inside the side pores, Figure 4.2B) or from the configuration where the ions are inside the central pore (Figure 4.1B). The results obtained from these different initial configurations do not differ from each other. The non-equilibrium simulations are conducted until the ions relocate inside the side pores and steady-states flow is achieved, as described in the 4.3 section. The results presented in Figure 4.7A indicate that in the side pores the velocity of water molecules is nonzero (thus consistent with hydrodynamic slip), and it undistinguishable from the velocity of the ions (suggesting that the ions move with water). The velocity of Na+ ions is larger than that of Cl- ions because of steric effects within the narrow side pores considered in my design. In the centre of the device water molecules flow with higher velocity than in the side pores because the pore is wider, as expected. The hydrodynamic slip observed both in the side pores and within the central pore is consistent with prior experimental and modelling observations [27, 60, 68-70]. My results suggest that the slippage of the compact EDLs observed within the composite cell can tremendously improve the operation of CD devices because CD is based on the physical adsorption of ions onto charged porous electrodes. In both flow-by [102, 115] and flow-through [140] processes when salty water is exposed to a pair of fresh electrodes the counter-ions adsorb onto the charged electrodes, and fresh water is produced. However, because once the ions enter the electrodes they remain trapped there, regeneration is necessary [102, 115, 140], and the process is not continuous, unless complex operations are designed (e.g., desalination with wires) [141]. As opposed to existing technologies, the composite cell I propose promises the possibility of continuous operation, because
there is no need of electrodes regeneration (Figure 4.7B). I christened my designed ‘continuous electrode-membrane desalination cell’.
Figure 4.7 Velocity of water molecules, Na+ and Cl- ions inside the composite electrode cell (A) as obtained from Poiseuille flow simulations. The water flows with the velocity of ~ 3.5m/s within the central pore. In the bottom side-pore, water and Na+ ions flow with the velocity of 1.9m/s. In the top side-pore, water and Cl- ions
flow with the velocity of 1.2m/s. Despite these differences in flow velocities, all the electrolyte solutions slip inside all of the pores in the composite cell, promising continuous CD operation (B).
The continuous electrode-membrane desalination cell I envision is operated as follows: the salty water is fed continuously into the central pore (note that in my non- equilibrium simulations, because of periodic boundary conditions, salty water cannot be fed to the system); during operation the ions diffuse from the central pore to the side pores because of the applied voltages. Compared to existing CD devices [142], the envisioned cell has the advantage that the two neutral graphene membranes provide a physical barrier to separate purified from salty water (central and side
the operation enabled by the proposed design differs substantially compared to flow- through, flow-by, and desalination-with-wires CD devices in which the ions, once trapped, remain immobilized inside the porous electrodes [115]. I estimate water permeability (30L/cm2/day/MPa, under the assumption that 10% of the cross surface area of an hypothetical membrane that embeds the proposed cell is constituted by pores) much larger than that obtained from current membrane-based water desalination techniques [143]. At optimum conditions (surface charge density =+/-4.2µC/cm2 and salinity < 10.9g/l) the proposed desalination cell can capture all salty ions within the side pores (100% rejection), recover 70% of the salty water initially fed to the system, and yield a charge efficiency of 83%. Higher charge efficiency, pershaps 100% can be obtained at higher salinity using larger pores (Figure 4.8). Note that 100% charge efficiency has been reported in the literature [115]. Operating the cell will require energy for applying voltages on the electrodes and pumping the salty water through the cell. Because fast water transport through graphitic nanopores and carbon nanotubes under small applied pressures was experimentally observed [60, 70], because it has been reported that CD of brackish water consumes much less energy than reverse osmosis does [102, 144, 145], and because high charge efficiency can be obtained in the engineered cell I propose, I expect the proposed cell to be competitive against existing processes. Unfortunately, quantification of operational costs cannot be conducted reliably at this stage.
Figure 4.8 Representative simulation snapshot at equilibrium for the composite electrochemical cells in which the side-pore size is H=10Å, salinity 18g/l, and surface charge density µ =+/-4.2µC/cm2 [6e/(5.4x4.2nm2)]. There are 6 ion pairs extracted to the side-pores. Unity charge efficiency is obtained: 6 charge units on each electrode extract 6 monovalent ions from NaCl solution.